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Wikipedia / Algebraic function

In mathematics, an algebraic function is a function that satisfies a polynomial equation are polynomials (not all zero). Basic examples of algebraic functions are polynomial functions, rational functions, the nth root), whose graph is the top half of the standard unit circle. Algebraic functions are contrasted with transcendental functions, such as the exponential function, logarithm, and the trigonometric functions.

Algebraic functions are usually treated more generally as multivalued functions in the same package. Algebraic functions are most often studied over the complex numbers is an irreducible polynomial and complex coefficients. can be expressed as having the two single-valued branches, with branch points. This particular function can be written using finitely many algebraic operations and extraction of nth roots, but this is not generally the case such as with the Bring radical. Over the complex numbers, algebraic functions have local holomorphic branches away from finitely many branch points and poles, and are naturally studied as meromorphic functions on compact Riemann surfaces.

More generally, over a field is defined algebraically as an element algebraic over the rational function field,

.

An variables is an element algebraic over the field of rational functions. Equivalently, it satisfies a polynomial equation

In one variable, algebraic functions are closely related to algebraic curves and their function fields; in the separable case, they may also be studied via finite or ramified covers of the projective line.

Algebraic functions in one variable

Basic examples

satisfies

satisfies

th root of a polynomial or rational function is algebraic, since it satisfies an equation such as

Many elementary algebraic functions can be obtained from rational functions by algebraic operations and extraction of roots. However, algebraic functions are more general than functions expressible by radicals. By Galois theory, roots of a general polynomial equation of degree five or higher cannot be expressed by radicals.

-line.

The role of complex numbers

From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field, such as zeros of the leading coefficient or of the discriminant, are responsible for poles, multiple roots, and branch points.

Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation

Using the cubic formula, we get

the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanying image.

It may be proven that there is no way to express this function in terms of nth roots using real numbers only, even though the resulting function is real-valued on the domain of the graph shown.

On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense.

Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 ∈ C is such that the polynomial p(x0, y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of x0. Choose a system of n non-overlapping discs Δi containing each of these zeros. Then by the argument principle

By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x, y) has only one root in Δi, given by the residue theorem:

which is an analytic function.

Branch points and Puiseux series

, the branches can be represented by convergent Puiseux series describes the ramification of the branch. Algebraic functions have no singularities other than poles and algebraic branch points.

Monodromy

Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x), provided that x is not a critical value or the discriminant vanishes. Hence there are only finitely many such values c1, ..., cm.

A close analysis of the properties of the function elements fi near the critical values can be used to show that the monodromy cover is ramified over the critical values (and possibly the point at infinity). Thus the holomorphic extension of the fi has at worst algebraic poles and ordinary algebraic branchings over the critical values.

Note that, away from the critical values, we have

since the fi are by definition the distinct zeros of p. Analytic continuation of the local branches around loops avoiding the critical values permutes the branches; these permutations form the monodromy group of the algebraic function. (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.)

is the splitting field, equivalently the Galois closure of the extension generated by one branch, then the Galois group. Under the correspondence between finite branched covers of the Riemann sphere, this Galois group is identified with the monodromy group of the covering. Thus the monodromy action realizes the Galois group of the splitting field as a permutation group on the branches.

Closure properties

, for instance by the resultant

implicitly gives

. By contrast, an antiderivative of an algebraic function need not be algebraic. Integrals of algebraic functions lead more generally to Abelian integrals, such as elliptic integrals.

History

The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in which he writes:

  1. let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms.

See also

References